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A New Definition of Convergence of a Double Sequence and a Double Series and Silverman-Toeplitz Theorem

P N Natarajan — 2014

Commentationes Mathematicae

Throughout this paper, entries of 4-dimensional infinite matrices, double sequences and double series are real or complex numbers. In the present paper, we introduce a new definition of convergence of a double sequence and a double series and record a few results on convergent double sequences. We also prove Silverman-Toeplitz theorem for double sequences and series.

Some Characterizations of Schur Matrices in Ultrametric Fields

P. N. Natarajan — 2012

Commentationes Mathematicae

In this short paper, K denotes a complete, non-trivially valued, ultrametric field. Sequences and infinite matrices have entries in K. We prove a few characterizations of Schur matrices in K . We then deduce some non-inclusion theorems modelled on the results of Agnew [1] and Fridy [3] in the classical case.

Cauchy Multiplication of Euler Summable Series in Ultrametric Fields

R. DeepaP N NatarajanV. Srinivasan — 2013

Commentationes Mathematicae

Euler summability method in a complete, non-trivially valued, ultrametric field of the characteristic zero was introduced by Natarajan in [7]. Some properties of the Euler summability method in such fields were studied in [2] and [7]. The purpose of the present note is to continue the study and to prove a pair of theorems on the Cauchy product of Euler summable sequences and series.

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